IMPULSE AND MOMENTUM 263 making an angle a with the plane, and rebounds time after time. Prove that 2 v sin a , i<2sin~c* where T is the total time of flight after the first impact, R the total range, and e the coefficient of restitution. 20. In the preceding problem find the values of T and R for the following special cases: (a) v = 500 meters per second, a = 30°, e = 0.5. (b) v = 500 meters per second, a. = 90°, e = 0.9. 21. A particle is projected horizontally from the top of a smooth inclined plane. Derive an expression for the time at the end of which the particle stops rebounding and slides down the plane. Compute its value for the following special cases: (a) VQ = 500 feet per second, a — 45°, e = 0.5. (b) v0 = 500 feet per second, a = 30°, e = 0.3. 22. In the preceding problem find the distance the particle moves along the plane before it stops rebounding. 23. In problem 21 find the velocity of the particle at the instant it stops rebounding. 24. A bead slides down a smooth circular wire, which is in a vertical plane, and strikes a similar bead at the lowest point of the wire. If during the collision the first bead conies to rest, show that the second bead will rise to a height erh and on its return will follow the first bead to a height 62 (i _ 6)2 /^ where h is the height from which the first bead falls. 25. Two equal spheres, which are in contact, move in a direction perpendicular to their line of centers and impinge simultaneously on a third equal sphere which is at rest. Supposing the contacts to be perfectly smooth and elastic find the velocity of each sphere after the collision. 26. A bullet hits and instantly kills a bird, while passing the highest point of its trajectory. Supposing the bullet to stay imbedded in the bird, and the bird to have been at rest when shot, find the distance between the place of firing and the point where the bird strikes the ground. 27. Two particles of masses mi and w2 are connected by an inextensible string of negligible mass. The second particle is placed on a smooth horizontal table while the first is allowed to fall from the edge of the table. When the falling particle reaches a distance h from the top of the table the string becomes tight. Find the velocity with which the second particle, begins to move.